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Geometry

Purpose of Course  showclose

“Everything is numbers.”  This phrase was uttered by the lead character, Dr. Charlie Epps, on the hit television show “NUMB3RS.”  If everything has a mathematical underpinning, then it follows that everything is somehow mathematically connected, even if it is only in some odd, “six degrees of separation (or Kevin Bacon)” kind of way.

Geometry is the study of space (for now, mainly two-dimensional, with some three-dimensional thrown in) and the relationships of objects contained inside.  It is one of the more relatable math courses, because it often answers that age-old question, “When am I ever going to use this in real life?”  Look around you right now.  Do you see any triangles?  Can you spot any circles?  Do you see any books that look like they are twice the size of other books?  Does your wall have paint on it?

In geometry, you will explore the objects that make up our universe.  Most people never give a second thought to how things are constructed, but there are geometric rules at play.  Most people never think twice about a rocket launch, but if that rocket is not launched at an exact angle, it will miss its target.  A football field has to be measured out to be a rectangle; if you used another shape, such as a trapezoid, that would give an unfair advantage to one team, because that one team would have more space to work with.

In this course, you will study the relationships between lines and angles.  Have you ever looked at a street map?  Believe it or not, there is a lot of geometry on a map, as you will see from this course.  You will learn to calculate how much space an object covers, which is useful if you ever have to, say, buy some paint.  You will learn to determine how much space is inside of a three-dimensional object, which is useful for those times you are trying to fit four suitcases, three kids, two adults, and a dog into the back of your vehicle.

These are just some of the topics you will be learning.  As you will quickly see, everything is not just numbers; it is also relationships.  Even nature itself knows this.  What did the little acorn say when it grew up?  “Gee, I’m a tree!”

Course Information  showclose

Welcome to RWM 103.  General information about the course and its requirements can be found below.

Course Designer: Eric Clark

Primary Resources: This course is composed of a range of different free, online materials.  However, the course makes primary use of the following materials:
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  Please pay special attention to Units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units.  You will also need to complete:
  • Subunit 1.1 Assessment
  • Sub-subunit 1.4.4 Assessment
  • Sub-subunit 1.4.5 Assessment
  • Sub-subunit 1.4.7 Assessment
  • Sub-subunit 2.2.5 Assessment
  • Subunit 3.1 Assessment
  • Sub-subunit 3.2.3 Assessment
  • Sub-subunit 3.3.5 Assessment
  • Sub-subunit 4.6.2 Assessment
  • Subunit 5.2 Assessment
  • Subunit 5.7 Assessment
  • Subunit 6.1 Assessment
  • Subunit 6.3 Assessment
  • Subunit 7.1 Assessment
  • Subunit 7.3 Assessment
  • Subunit 7.4 Assessment
  • Subunit 9.2 Assessment
  • Subunit 9.3 Assessment
  • Subunit 9.6 Assessment
  • Subunit 9.10 Assessment
  • Subunit 10.7 Assessment
  • The Final Exam
Please note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the assessments listed above as well as all resources in each unit.

In order to pass this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: This course should take you approximately a total of 144.75 hours to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  It may be useful to take a look at these time advisories and determine how much time you have over the next few weeks to complete each unit and to then set goals for yourself.  For example, Unit 1 should take you 17.5 hours.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 3 hours) on Monday night; subunits 1.2 and 1.3 (a total of 2.25 hours) on Tuesday night; etc.

Tips/Suggestions: Please make sure to take comprehensive notes as you work through each resource.  These notes will serve as a useful review as you study for your Final Exam.  Complete all practice problems, because practice will allow you to fully understand each concept.

Khan Academy  
This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes  showclose

NOTE: Each of the learning outcomes listed below has been aligned with one or more of the Common Core standards in mathematics. This alignment is reflected in the numbered notation listed alongside each outcome below. For more information on Common Core standards, please read here.

Upon successful completion of this course, the student will be able to:
  • Define and identify angles, rays, line segments, and points, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (HSG-CO.A.1)
  • Use the definition of congruence to decide if figures are congruent. (HSG-CO.B.6)
  • Use the definition of congruence to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. (HSG-CO.B.7)
  • Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence. (HSG-CO.B.8)
  • Given two figures, decide if they are similar. (HSG-SRT.A.2)
  • Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (HSG-SRT.A.3)
  • Explain and use the relationship between the sine and cosine of complementary angles. (HSG-SRT.C.7)
  • Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (HSG-SRT.C.8)
  • Identify and describe relationships among inscribed angles, radii, and chords. (HSG-C.A.2)
  • Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.(HSG-GMD.A.3)
  • Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. (HSG-GMD.B.4)
  • Use geometric shapes, their measures, and their properties to describe objects. (HSG-MG.A.1)
  • Recall the formulas for the area and circumference of a circle and use them to solve problems. (7.G.B.4)
  • Using facts about supplementary, complementary, vertical, and adjacent angles, write and solve simple equations for an unknown angle in a figure. (7.G.B.5)
  • Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.B.6) 

Course Requirements  showclose

In order to take this course, you must:

√    have access to a computer.

√    have continuous broadband Internet access.

√    have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash).

√    have the ability to download and save files and documents to a computer.

√    have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    have competency in the English language.

√    have read the Saylor Student Handbook.

√    have completed courses RWM101: Real World Math I: Foundations and RWM102: Real World Math II: Algebra.

Preliminary Information

Unit Outline show close


Expand All Resources Collapse All Resources
  • Unit 1: Basics of Geometry  

    From a young age, we learn basic vocabulary and basic concepts that lead us to understand greater concepts.  In this unit, you will learn about the basic building blocks of Geometry.

    Everything has a proper name.  We often seem to use nicknames: “fridge” instead of “refrigerator,” “phone” instead of “telephone,” and so on.  Still, we know what the proper terminology is, even if we do not use it as often as we probably should.  Knowing the meanings of several basic terms in Geometry is important, because these terms are going to keep popping up over and over again.  Whether it is knowing a simple definition, or understanding the relationships between angles, these basic items are a “must know” in Geometry.

    In life, we also classify things.  We tend to classify by height, age, weight, food preference, nationality, etc.  We will use pretty much anything we can think of for classification purposes.  While some classifications might seem unnecessary to us, others seem quite important.  Classifying berries, for example, can help you to keep from eating a poisonous one that would make you sick.  Without classification, we would not have a way to keep the good ones separate from the bad ones.  The same holds true in Geometry.  Classifying shapes like triangles can help us recognize these shapes faster and eventually recognize the rules and relationships pertaining to their classification.

    Time Advisory   show close
    Learning Outcomes   show close
  • 1.1 Language and Notation of Basic Geometry  
  • 1.2 Distance between Two Points  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance between Two Points”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance between Two Points” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the embedded videos to learn about the distance between two points and the segment addition postulate.  Watch the presentations carefully two or three times until you are able to explain how apply finding the distance between two points and the segment addition postulate.  Then, read the following sections: “Guidance” (omitting the Distance between Two Points B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of measuring on a ruler and finding the distance between points.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Watching these videos, reading the material, and pausing to take notes should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance between Two Points”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance between Two Points” (HTML)

      Instructions: Please click on the link above, select the “Exercises” tab, and complete the activity on the distance between two points and the angle addition postulate.  Compute the answer to the given problem, and select your response.  Click on “Submit Answer” to see if you choose the correct or an incorrect answer.  Then, click on “Next” to move on to another question.  You have completed the activity when you see “You have reached the maximum practice questions.”  (Note: You can refresh the page if you would like additional practice problems and the questions will be different than the first set.)  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.3 Midpoints and Segment Bisectors  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Midpoints and Segment Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Midpoints and Segment Bisectors” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first two videos (5 minutes and 7 minutes, respectively) to learn about midpoints and segment bisectors.  You do not need to watch the “James Sousa: Segment Midpoint and Segment Perpendicular Bisector” video.  Watch the presentations carefully two or three times until you are able to explain how apply the segment addition postulate.  Then, read the following sections: “Guidance” (omitting the Midpoints and Segment Bisectors B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of measuring on a ruler and finding the distance between points.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Watching this lecture, reading the material, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Midpoints and Segment Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Midpoints and Segment Bisectors” (HTML and YouTube)

      Instructions: Please click on the link above, select the “Exercises” tab, and complete the activity on the midpoints and segment bisectors.  Compute the answer to the given problem, and select your response.  Click on “Submit Answer” to see if you choose the correct or an incorrect answer.  Then, click on “Next” to move on to another question.  You have completed the activity when you see “You have reached the maximum practice questions.” (Note: You can refresh the page if you would like additional practice problems and the questions will be different than the first set.)  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.4 Angles  
  • 1.4.1 Angle Basics  
  • 1.4.2 Measuring Angles  
  • 1.4.3 Congruent Angles and Angle Bisectors  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Angles and Angle Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Angles and Angle Bisectors” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first two videos (approximately 5 minutes and 3 minutes, respectively) to learn about angle bisectors.  You do not need to watch the “James Sousa: Bisector and Midpoint Exercises” video.  Watch the presentation carefully two or three times until you are able to understand problems with an angle bisector.  Then, read the following sections: “Guidance” (omitting Congruent Angles and Angle Bisectors B video), “Vocabulary,” and “Guided Practice.”  This section provides examples identifying congruent angles and applying properties of an angle bisector.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Watching this lecture, reading the sections, and pausing to take notes should take no more than 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Angles and Angle Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Angles and Angle Bisectors” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–10.  These exercises will provide you with the opportunity to use properties of congruent angles and angle bisectors.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Congruent Angles and Angle Bisectors Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.4.4 Acute, Obtuse, and Right Angles  
  • 1.4.5 Complementary and Supplementary Angles  
    • Lecture: Khan Academy’s “Complementary and Supplementary Angles” and “Identifying Complementary and Supplementary Angles”

      Link: Khan Academy’s “Complementary and Supplementary Angles” (YouTube) and “Identifying Complementary and Supplementary Angles” (YouTube)

      Instructions: Please click on the links above, and take notes as you watch these videos to learn how to identify supplementary and complementary angles.  Watch the first presentation carefully to understand the basics of complementary, supplementary, adjacent, and straight angles.  Watch the second presentations carefully two or three times until you are able to identify types of angles.

      Watching these lectures and pausing to take notes should take no more than 1 hour.

      Terms of Use: These videos are licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Complementary Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Complementary Angles” (HTML)

      Instructions: Please click on the link above, select the “Exercises” tab, and complete the problems on complementary angles.  Compute the answer to the given problem, and select your response.  Then, click “Submit Answer” to see if you chose the correct answer or an incorrect answer.  Select “Next” to move on to another question.  You have completed the activity when you see “You have reached the maximum practice questions.” (Note: You can refresh the page if you would like additional practice problems and the questions will be different than the first set.)  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Supplementary Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Supplementary Angles” (HTML)

      Instructions: Please click on the link above, select the “Exercises” tab, and complete the problems on supplementary angles.  Compute the answer to the given problem, and select your response.  Click on “Submit Answer” to see if you chose the correct answer or an incorrect answer.  Select “Next” to move on to another question.  You have completed the activity when you see “You have reached the maximum practice questions.” (Note: You can refresh the page if you would like additional practice problems and the questions will be different than the first set.)  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this interactive lab should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Assessment: Khan Academy’s “Complementary and Supplementary Angles”

      Link: Khan Academy’s “Complementary and Supplementary Angles” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on complementary and supplementary angles.  You can review the concepts associated with these questions with the Khan videos in sub-subunit 1.4.5.  Determine the value for the problem, and enter your answer in the answer box.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy. 

      See a broken link? Please let us know!

  • 1.4.6 Linear Pairs  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Linear Pairs”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Linear Pairs” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about linear pairs.  You do not need to watch the “James Sousa: Linear Pairs” video.  Watch the presentation carefully two or three times until you are able to understand problems with linear pairs.  Then, read the following sections: “Guidance” (omitting Linear Pairs B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of adjacent angles and angles that form a linear pair.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Watching this video, reading the material, and pausing to take notes will take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Linear Pairs”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Linear Pairs” (HTML and YouTube)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–16.  These exercises will provide you with the opportunity to determine angles using linear pairs.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Linear Pairs Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.4.7 Vertical Angles  
    • Lecture: Khan Academy’s “Angles at the Intersection of Two Lines”

      Link: Khan Academy’s “Angles at the Intersection of Two Lines” (YouTube)

      Instructions: Please click on the link above and take notes as you watch this video to learn how vertical angles are equal to each other.  Watch the presentation carefully two or three times to understand how vertical angles are equal.

      Watching these lectures and pausing to take notes should take approximately 30 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Vertical Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Vertical Angles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video (approximately 3 minutes) to learn about linear pairs.  You do not need to watch the “James Sousa: Vertical Angles” video.  Watch the presentation carefully two or three times until you are able to understand problems with vertical angles.  Then, read the following sections: “Guidance” (omitting Vertical Angles B video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of vertical angles and the vertical angles theorem.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and pausing to take notes should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Vertical Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Vertical Angles” (HTML and YouTube)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to determine angles using vertical angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Vertical Angles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: Khan Academy’s “Vertical Angles 2”

      Link: Khan Academy’s “Vertical Angles 2” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on vertical angles.  You can review the concepts associated with these questions with the Khan videos in sub-subunit 1.4.7.  Determine the value for the problem and enter your answer in the answer box.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy. 

      See a broken link? Please let us know!

  • 1.5 Polygon Classification  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Classifying Polygons”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Classifying Polygons” (HTML)

      Instructions: Please click on the link above and study the “Points, Lines, and Planes” section (Pages 49–54 in the textbook), stopping at “Using the Distance Formula on Polygons.”  This section provides examples of various polygons and concave and convex polygons.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading this section should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Classifying Polygons”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Classifying Polygons” (HTML)

      Instructions: Please click on the link above and complete questions 1–10 (“Lesson Exercises” on Pages 55 and 56 in the textbook).  These questions will provide you with the opportunity to identify polygons and apply polygon properties.  The solutions to these problems are located in the “Answers” section on page 56 of the textbook.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this activity should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • Unit 2: Parallel and Perpendicular Lines  

    Simply put, parallel lines are lines in a plane that do not intersect, while perpendicular lines are lines that intersect at a right angle (90°).  Parallel and perpendicular lines are all around us.  For example, railroad tracks, iron fencing, latticework, and kitchen tables all have parallel and/or perpendicular lines.

    Have you ever looked at a tile floor that has an intricate pattern of crisscrossing lines?  The pattern has to be carefully replicated, following special rules.  When parallel lines cross other lines, they create their own set of special rules, which may be used in making these patterns.  In this unit, you will learn what happens when parallel lines cross other lines and how all the created angles relate to each other.

    Time Advisory   show close
    Learning Outcomes   show close
  • 2.1 Lines  
  • 2.1.1 Parallel, Perpendicular, and Skew Lines  
  • 2.1.2 Angles and Transversals  
  • 2.2 Angles and Lines  
  • 2.2.1 Corresponding Angles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Corresponding Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Corresponding Angles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about corresponding angles.  You do not need to watch the “James Sousa: Angles and Transversals,” “Properties of Parallel Lines,” and “Corresponding Angles Converse” videos.  Watch the presentation carefully two or three times until you are able to understand problems with corresponding angles.  Then, read the following sections: “Guidance” (including Corresponding Angles B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of corresponding angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and taking notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Corresponding Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Corresponding Angles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to determine angles using corresponding angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account. Once you have completed the exercises, check your answers against the Saylor Foundation’s “Corresponding Angles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 2.2.2 Alternate Interior Angles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Interior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Interior Angles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about alternate interior angles.  You do not need to watch the “James Sousa: Angles and Transversals,” “Proof that Alternate Interior Angles are Congruent,” or “Proof of Alternate Interior Angles Converse” videos.  Watch the presentation carefully two or three times until you are able to understand problems with alternate interior angles.  Then, read the following sections: “Guidance” (including Alternate Interior Angles B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of corresponding angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and taking notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Interior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Interior Angles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to determine angles using alternate interior angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Alternate Interior Angles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 2.2.3 Alternate Exterior Angles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Exterior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Exterior Angles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about alternate exterior angles.  You do not need to watch the “James Sousa: Angles and Transversals” and “Proof of Alternate Exterior Angles Converse” videos.  Watch the presentation carefully two or three times until you are able to understand problems with alternate exterior angles.  Then, read the following sections: “Guidance” (including Alternate Exterior Angles B video), “Vocabulary,” and “Guided Practice.”  This section provides examples of alternate exterior angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Exterior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Alternate Exterior Angles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to determine angles using alternate exterior angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Alternate Exterior Angles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 2.2.4 Same Side Interior Angles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Same Side Interior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Same Side Interior Angles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about same side interior angles.  You do not need to watch the “James Sousa: Angles and Transversals,” “Proof that Consecutive Interior Angles Are Supplementary,” and “Proof of Consecutive Interior Angles Converse” videos.  Watch the presentation carefully two or three times until you are able to understand problems with same side interior angles.  Then, read the following sections: “Guidance,” “Vocabulary” and “Guided Practice.”  This section provides examples of same side interior angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Same Side Interior Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Same Side Interior Angles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to determine angles using alternate exterior angles.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Same Side Interior Angles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 2.2.5 Angle Game  
  • 2.3 Distance Formula in the Coordinate Plane  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance Formula in the Coordinate Plane”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance Formula in the Coordinate Plane” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about find the distance between two points in the coordinate plane.  You do not need to watch the “James Sousa: The Distance Formula” video.  Watch the presentation carefully two or three times until you are able to understand problems finding the distance between two points.  Then, read the following sections: “Guidance” (omitting Example C and the Distance Formula in the Coordinate Plane B video), “Vocabulary,” and “Guided Practice,” stopping at “Guide Practice” question 3.  This section provides examples of find the distance between two points in the coordinate plane.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and pausing to take notes should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance Formula in the Coordinate Plane”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Distance Formula in the Coordinate Plane” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–8 and 13.  These exercises will provide you with the opportunity to find the distance between two points in the coordinate plane.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Distance Formula in the Coordinate Plane Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • Unit 3: Triangles, Congruence, and Other Relationships  

    Making sure that objects are of equal size is important in life.  Whether pieces of furniture, automobiles, or just pieces of a candy bar shared between siblings, we often have to make sure objects are the same size and shape.

    Have you ever had to prove something to a child?  Sometimes, the best way to do that is to give the child a series of logical statements and then show how it all summarizes to one conclusion.  In this unit, you will focus on showing that two triangles are identical.  There are different methods to doing this, and the one you use will depend upon the information you have available to you.  It is nothing more than following a series of logical statements and then showing how the statements draw a conclusion.

    In this unit, you will also learn more about triangles and their various parts.  Triangles, believe it or not, are used more often in life than you might think.  Triangles are often used to estimate distances.  The process of triangulation has been used for over two thousand years; with this technique, we can use two known locations and determine the distance to a location that we can see but cannot necessarily get to, like a boat out in a lake.  Today, we use triangulation for navigating boats, surveying land, launching model rockets, and other activities.

    We can also use triangles to map things out.  For example, city planners often use the circumcenter of a triangle, which is a point in the middle that is equally distant from all three sides.  They might use this to determine the location of a major parking garage, so that it is the same distance from three different companies, or they might want to make sure that a city monument is in the middle of a town plaza.

    If you live in a house, triangles are right over your heads.  Construction workers use the midsegment of a triangle to help strengthen roof trusses when they build.  If the truss is not supported properly, it could collapse, leading to lots of problems for the poor homeowner.

    Time Advisory   show close
    Learning Outcomes   show close
  • 3.1 Triangle Classification  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Classification” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about classifying triangles based on angles and sides.  You do not need to watch the “James Sousa: Types of Triangles” video.  Watch the presentation carefully two or three times until you are able to understand problems classifying triangles.  Then, read the following sections: “Guidance” (omitting “Triangle Classification B” video), “Vocabulary,” and “Guided Practice.”  This section provides examples of triangle classification.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Classification” (HTML)

      Instructions: Please click on the link above, select the “Exercises” tab, and complete the problems on classifying triangles.  Determine the answer to the given problem, and select your response.  Click on “Submit Answer” to see if the answer you chose is correct or incorrect.  Select “Next” to move on to another question.  You have completed the activity when you see “You have reached the maximum practice questions.”  (Note: You can refresh the page if you would like additional practice problems and the questions will be different than the first set.)  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Assessment: Khan Academy’s “Triangle Types”

      Link: Khan Academy’s “Triangle Types” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on vertical angles.  You can review the concepts associated with these questions with the material in subunit 3.1.  Select the correct answer choice for each problem.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 30 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy. 

      See a broken link? Please let us know!

  • 3.2 Angle Properties of Triangles  
  • 3.2.1 Triangle Sum Theorem  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Sum Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Sum Theorem” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first two videos to learn about the sum of the interior angles of a triangle.  You do not need to watch the “James Sousa: Proving the Triangle Sum Theorem” video.  Watch the presentations carefully two or three times until you are able to understand why the angles of a triangle add up to 180 degrees.  Then, read the following sections: “Guidance” (omitting the two-column proof of the Triangle Sum, but watching “Chapter 4 Triangle Sum Theorem B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding angles within a triangle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Sum Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Sum Theorem” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to find the missing angle(s) of triangles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Triangle Sum Theorem Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 3.2.2 Exterior Angle Theorem  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Exterior Angle Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Exterior Angle Theorem” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first two videos to learn about the exterior angles and the relationship with triangles.  You do not need to watch the “James Sousa: Introduction to the Exterior Angles of a Triangle” and “Proof of the Exterior Angles Theorem” videos.  Watch the presentations carefully two or three times until you are able to understand why the exterior angles add up to 360 degrees.  Then, read the “Guidance” (omitting the two-column proof of the Exterior Angle Theorem, and “Chapter 4 Exterior Angles Theorem B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of how to find the missing interior and exterior angles of a triangle.

      Reading the material, watching the videos, and pausing to take notes will take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Exterior Angle Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Exterior Angle Theorem” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–12.  These exercises will provide you with the opportunity to find the missing angles of triangles as well as exterior angles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Exterior Angle Theorem Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 3.2.3 Triangle Examples  

    Note: The material from sub-subunits 1.4.5–1.4.7, 2.4, 3.2.1, and 3.2.2 is applicable to the problems within this sub-subunit.  If needed, please refer back to these sections of the course for review.

  • 3.3 Triangle Congruence  
  • 3.3.1 Congruent Figures  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Figures”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Figures” (HTML)

      Instructions: Please click on the link above and study the “Congruent Figures” section (Pages 206–212 in the textbook).  This section provides discussions on defining and writing congruence statements in triangles, the third angle theorem, and congruence properties.

      Reading this section should take approximately 30 minutes.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Figures”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Congruent Figures” (HTML)

      Instructions: Please click on the link above and complete questionss 1–10 (“Lesson Exercises” Pages 212 and 213 in the textbook).  These exercises will provide you with the opportunity to identify congruent figures and explain why certain triangles are congruent.  The solutions to these problems are located in the “Answers” section (on page 213 of the textbook).

      This activity should take approximately 30 minutes.

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  • 3.3.2 Congruent Triangles and SSS  
  • 3.3.3 Other Triangle Congruent Postulates  
  • 3.3.4 More on Why SSA Is Not a Postulate  
  • 3.3.5 Finding Congruent Triangles  
  • 3.4 Isosceles Triangles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Isosceles Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Isosceles Triangles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first videos to learn about the isosceles triangles angles and their properties.  You do not need to watch the “James Sousa: How to Construct an Isosceles Triangle” and “Proof of Isosceles Triangle Theorem” videos.  Then, take notes as you watch the last video on “Using the Properties of Isosceles Triangle to Determine Values” (approximately 4 minutes) to learn about the isosceles triangles applications.  Watch the presentations carefully two or three times until you are able to understand the properties of an isosceles triangle.  Then, read the “Guidance” (omitting the two-column proof of the Base Angles Theorem and “Chapter 4 Isosceles Triangle B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of how to find the missing sides and angles of an isosceles triangle.

      Reading the material, watching the video, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Lecture: Khan Academy’s “Another Isosceles Example Problem”

      Link: Khan Academy’s “Another Isosceles Example Problem” (YouTube)

      Instructions: Please click on the link above and take notes as you watch this video to determine the missing sides and angles of an isosceles triangle.  Watch the presentation carefully two or three times to understand on how to write algebraic equations to find the missing angles and sides of an isosceles triangle.

      Watching this lecture and pausing to take notes should take approximately 30 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Isosceles Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Isosceles Triangles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–9.  These exercises will provide you with the opportunity to find the missing angles and sides of isosceles triangles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Isosceles Triangles Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 3.5 Equilateral Triangles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Equilateral Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Equilateral Triangles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first videos (approximately 4 minutes) to learn about the equilateral triangles angles and their properties.  You do not need to watch the “James Sousa: How to Construct an Equilateral Triangle” and “Proof of Equilateral Triangle Theorem” videos.  Then, take notes as you watch the last video on “Using the Properties of Equilateral Triangles” (approximately 4 minutes) to learn about the applications of equilateral triangles.  Watch the presentations carefully two or three times until you are able to understand the properties of an isosceles triangle.  Then, read the “Guidance” (omitting “Chapter 4 Equilateral Triangle B” video), “Vocabulary,” and “Guided Practice” (omitting Guided Practice #2) sections.  This section provides examples of find the missing sides and angles of an equilateral triangle.

      Reading the material, watching the video, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Equilateral Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Equilateral Triangles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to find the missing angles and sides of isosceles triangles.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Equilateral Triangles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • Unit 4: Triangle Relationships  

    In this unit, you will learn more about triangles and their various parts.  Triangles, believe it or not, are used more often in life than you might think.  Triangles are often used to estimate distances.  The process of triangulation has been used for over two thousand years; with this technique, we can use two known locations and determine the distance to a location that we can see but cannot necessarily get to, like a boat out in a lake.  Today, we use triangulation for navigating boats, surveying land, launching model rockets, and other activities.

    We can also use triangles to map things out.  For example, city planners often use the circumcenter of a triangle, which is a point in the middle that is equally distant from all three sides.  They might use this to determine the location of a major parking garage, so that it is the same distance from three different companies, or they might want to make sure that a city monument is in the middle of a town plaza.

    If you live in a house, triangles are right over your heads.  Construction workers use the midsegment of a triangle to help strengthen roof trusses when they build.  If the truss is not supported properly, it could collapse, leading to lots of problems for the poor homeowner.

    Time Advisory   show close
    Learning Outcomes   show close
  • 4.1 Midsegment Theorem  
  • 4.2 Perpendicular Bisectors  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Perpendicular Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Perpendicular Bisectors” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about perpendicular bisectors and their properties.  You do not need to watch the “James Sousa: Constructing Perpendicular Bisectors” and “Proof of Perpendicular Bisector Theorem” videos.  Then, take notes as you watch the last video on “Determine Values Using Perpendicular Bisectors” to learn about the perpendicular bisector applications.  Watch the presentation carefully two or three times until you are able to understand the properties of a perpendicular bisector in a triangle.  Then, read the “Guidance” (omitting “Proof of the Perpendicular Bisector Theorem Converse,” and “Chapter 5 Perpendicular Bisectors B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of find the missing sides and angles of an equilateral triangle.

      Reading the material, watching the video, and taking notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Perpendicular Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Perpendicular Bisectors” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1, 2, and 8–13.  These exercises will provide you with the opportunity to apply the properties of perpendicular bisectors.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Perpendicular Bisectors Solutions” (PDF)

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 4.3 Angle Bisectors in Triangles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Angle Bisectors in Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Angle Bisectors in Triangles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about angle bisectors in triangles.  You do not need to watch the “James Sousa: Introduction to Angle Bisectors,” “Proof of Angle Bisector Theorem,” and “Proof of the Angle Bisector Converse Theorem” videos.  Then, take notes as you watch the last video on “Solving for Unknown Values Using Angle Bisectors” to learn about the angle bisector applications.  Watch the presentation carefully two or three times until you are able to understand the properties of angle bisectors in a triangle.  Then, read the “Guidance” (omitting “Proof of the Angle Bisector Theorem” and the “Chapter 5 Angle Bisectors B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of how to find the missing angles by using properties of an angle bisector.

      Reading the material, watching the video, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Angle Bisectors in Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Angle Bisectors in Triangles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–8 and 10–13.  These exercises will provide you with the opportunity to apply the properties of angle bisectors.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Angle Bisectors in Triangles Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 4.4 Medians  
  • 4.5 Altitudes  
  • 4.6 Triangle Inequalities  
  • 4.6.1 Triangle Inequality Theorem  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Inequality Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Inequality Theorem” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the triangle inequality theorem.  Watch the presentation carefully two or three times until you are able to understand and apply the triangle inequality theorem.  Then, read the “Guidance” (omitting the “Chapter 5 Triangle Inequality Theorem B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides applications of the triangle inequality theorem, which states that any side of a triangle must be equal to or less than the sum of the other two sides in order to form the triangle.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Inequality Theorem”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Triangle Inequality Theorem” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–16.  These exercises will provide you with the opportunity to apply the triangle inequality theorem.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Triangle Inequality Theorem Solutions” (PDF).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 4.6.2 Inequalities in Two Triangles  
  • Unit 5: Polygons and Quadrilaterals  

    Polygons are everywhere around us.  If you see a shape that has no curves, then it is a polygon.  As a child, you probably learned the names of many common polygons.  In this unit, you will focus on quadrilaterals, which are polygons with four sides.  They each have their own set of characteristics and share some characteristics with other quadrilaterals.  You will learn to recognize these characteristics and to use them to help you answer questions about quadrilaterals.

    Time Advisory   show close
    Learning Outcomes   show close
  • 5.1 Sum of Interior Angles of a Polygon  
  • 5.2 Sum of Exterior Angles of a Convex Polygon  
  • 5.3 Parallelograms  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelograms”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelograms” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the properties of a parallelogram.  Watch the presentations carefully two or three times until you are able to understand and apply the properties of a parallelogram.  Then, read the “Guidance” (omitting “Proof of Opposite Theorems”), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of applying parallelogram properties.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelograms”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelograms” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–23 and 26–29.  These exercises will provide you with the opportunity to apply the parallelogram properties.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Parallelograms Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

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  • 5.4 Parallelogram Classification  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelogram Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelogram Classification” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about classifying other parallelogram and their properties.  Watch the presentations carefully two or three times until you are able to understand these properties of other parallelograms.  Then, read the “Guidance” (omitting “Example C”), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of applying parallelogram properties.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelogram Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Parallelogram Classification” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–19.  These exercises will provide you with the opportunity to apply the parallelogram properties.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Parallelogram Classification Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 5.5 Trapezoids  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Trapezoids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Trapezoids” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about trapezoids and their properties.  Watch the presentations carefully two or three times until you are able to understand these properties of trapezoids.  Then, read the “Guidance” (omitting “Example B”), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of applying trapezoid properties.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Trapezoids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Trapezoids” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–10 as well as 14 and 15.  These exercises will provide you with the opportunity to apply the parallelogram properties.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Trapezoids Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 5.6 Kites  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Kites”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Kites” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about kites and their properties.  Watch the presentations carefully two or three times until you are able to understand these properties of kites.  Then, read the “Guidance” (omitting “Proof”), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of applying kite properties.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Kites”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Kites” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–11, as well as 14 and 15.  These exercises will provide you with the opportunity to apply the kite properties.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Kites Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 5.7 Quadrilateral Classification  
  • 5.7.1 Coordinate Geometry  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Quadrilateral Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Quadrilateral Classification” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about classifying quadrilaterals using coordinate geometry.  Watch the presentations carefully two or three times until you are able to understand these properties of quadrilaterals.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of applying these properties to determine the quadrilateral.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour and 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Quadrilateral Classification”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Quadrilateral Classification” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to apply the properties of quadrilaterals.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Quadrilateral Classification Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

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  • 5.7.2 Quadrilateral Overview  
  • Unit 6: Similarity  

    In this unit, you will review ratios and proportions, and you will learn how they connect to similarity.  In the real world, you may see an item and think that it is similar to something else, whether it is because of the design, the size, or some other factors.  In Geometry, however, similarity has very strict rules attached to it.  You will learn to apply these rules to determine if two shapes, especially triangles, are similar.

    Time Advisory   show close
    Learning Outcomes   show close
  • 6.1 Similar Polygons  
  • 6.2 Similarity Postulates  
  • 6.2.1 Angle-Angle (AA) Similarity and Indirect Measurement  
  • 6.2.2 Side-Side-Side (SSS) Similarity and Side-Angle-Side (SAS) Similarity  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Similarity by SSS and SAS”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Similarity by SSS and SAS” (HTML)

      Instructions: Please click on the link above and study the “Similarity by SSS and SAS” section, (Pages 431–434 in the textbook), stopping at “Review Questions.”  This section provides examples of how to prove triangles are similar by SSS and SAS and how to find missing parts of similar triangles.

      Reading this section should take approximately 15 minutes.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Similarity by SSS and SAS”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Similarity by SSS and SAS” (HTML)

      Instructions: Please click on the link above and complete questions 1–11, (“Lesson Exercises” Pages 434 and 435 in the textbook).  These questions will provide you with the opportunity to apply SSS and SAS similarity.  The solutions to these problems are located in the “Answers” section (starting on page 435 of the textbook).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

      Submit Materials

  • 6.3 Triangle Proportionality and Parallel Lines and Transversals  
  • 6.4 Proportions with Angle Bisectors  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Proportions with Angle Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Proportions with Angle Bisectors” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about proportions with angle bisectors in triangles.  You do not need to watch the “James Sousa: Triangle Angle Bisector Theorem” video.  Then, take notes as you watch the last video on “Triangle Angle Bisector Theorem” to learn about the angle bisector applications.  Watch the presentation carefully two or three times until you are able to understand the properties of the triangle angle bisector theorem.  Then, read the “Guidance” (omitting “Chapter 7 Proportions with Angle Bisectors B” video), “Vocabulary,” and “Guided Practice” sections.  This section provides examples of how to find the missing angles using properties of an angle bisector.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Proportions with Angle Bisectors”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Proportions with Angle Bisectors” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–4 and 6–19.  These exercises will provide you with the opportunity to apply the properties of the triangle angle bisectors theorem.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Proportions with Angle Bisectors Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • Unit 7: Right Triangle Trigonometry  

    In this unit, you will learn the basics of trigonometry.  Trigonometry is used for several different measuring techniques.  You can determine the height of a building, for example, if you know how far away you are standing and at what angle your eyes form when looking up to its top.  In previous centuries, mariners could use trigonometry to help them set their course, using the heavens as a guide.

    Time Advisory   show close
    Learning Outcomes   show close
  • 7.1 Pythagorean Theorem  
  • 7.2 45-45-90 Triangles  
    • Lecture: Khan Academy’s “45-45-90 Triangles”

      Link: Khan Academy’s “45-45-90 Triangles” (YouTube)

      Instructions: Please click on the link above and take notes as you watch this video to understand the relationship between the angles and sides of a 45-45-90 triangle.  Watch this presentation carefully two or three times to understand how to find missing sides using this triangle.

      Watching this lecture and pausing to take notes should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “45-45-90-Right Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “45-45-90-Right Triangles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about 45-45-90 right triangles.  You do not need to watch the “James Sousa: Trigonometric Function Values of Special Angles” and “Solving Special Right Triangles” videos.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of problems using 45-45-90 triangles.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “45-45-90 Right Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “45-45-90 Right Triangles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to apply the relationship between the sides of a 45-45-90 right triangle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “45-45-90 Right Triangles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 7.3 30-60-90 Right Triangles  
    • Lecture: Khan Academy’s “30-60-90 Triangles II”

      Link: Khan Academy’s “30-60-90 Triangles II” (YouTube)

      Instructions: Please click on the link above and take notes as you watch this video to understand the relationship between the angles and sides of a 30-60-90 triangle.  Watch this presentation carefully two or three times to understand how to find missing sides by applying the properties of this triangle.

      Watching this lecture and pausing to take notes should take approximately 30 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “30-60-90-Right Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “30-60-90-Right Triangles” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about 30-60-90 right triangles.  You do not need to watch the “James Sousa: Trigonometric Function Values of Special Angles” and “Solving Special Right Triangles” videos.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of problems using 30-60-90 triangles.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “30-60-90-Right Triangles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “30-60-90-Right Triangles” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–13.  These exercises will provide you with the opportunity to apply the relationship between the sides of a 30-60-90 right triangle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “30-60-90 Right Triangles Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Assessment: Khan Academy’s “Special Right Triangles”

      Link: Khan Academy’s “Special Right Triangles” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on finding the missing sides of special right triangles.  You can review the concepts associated with these questions with the Khan videos from subunits 7.2 and 7.3.  Determine the value for the problem, and enter your answer in the answer box.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

  • 7.4 Sine, Cosine, and Tangent  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Sine, Cosine, and Tangent”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Sine, Cosine, and Tangent” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about trigonometric ratios.    Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of problems using these trigonometric ratios in right triangles.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour and 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Sine, Cosine, and Tangent”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Sine, Cosine, and Tangent” (HTML)
       
      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–16.  These exercises will provide you with the opportunity to apply the relationships of various trigonometric ratios.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Sine, Cosine, and Tangent Solutions” (PDF).
       
      Completing this activity should take approximately 45 minutes.
       
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      See a broken link? Please let us know!

    • Assessment: Khan Academy’s “Trigonometry 0.5”

      Link: Khan Academy’s “Trigonometry 0.5” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on basic trigonometric ratios.  You can review the concepts associated with these questions from subunit 7.4.  Determine the value for the problem, and enter your answer in the answer box.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

  • 7.5 Trigonometric Ratios with a Calculator  
  • Unit 8: Circles  

    Circles are everywhere you look, for example, tires, pools, fans, and watches.  You can probably think of several more items that are circular.  In this unit, you will learn about circles and arcs, which are slices of the edge of a circle.

    Believe it or not, arcs are used a lot in real life.  The feet of a rocking chair are arcs; the crust on a slice of pizza is an arc.  Anything that looks like an incomplete circle is an arc.  How are these arcs made?  What kinds are there?  How can we measure them?  These are questions that you will explore in this unit.

    Time Advisory   show close
    Learning Outcomes   show close
  • 8.1 Language and Notation of a Circle  
  • 8.2 Tangent Lines  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Tangent Lines”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Tangent Lines” (HTML)

      Instructions: Please click on the link above and study the “Tangent Lines” section (Pages 536–543 in the textbook) omitting the “Proof “ (Pages 537, 538, 540, and 541), stopping at “Proof” on page 543.  This section provides examples of how to identify tangent lines and its properties.

      Reading this section should take approximately 15 minutes.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Tangent Lines”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Tangent Lines” (HTML)

      Instructions: Please click on the link above and complete questions 1–8 (“Lesson Exercises” Pages 545–550 in the textbook).  These questions will provide you with the opportunity to apply properties of tangent lines and circles.  The solutions to these problems are located in the “Answers” section (starting on page 550 of the textbook).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 8.3 Arcs in Circles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arcs in a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arcs in a Circle” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the first video to learn about arcs in a circle.  You do not need to watch the “Central Angles” video.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of recognizing arcs of a circle and applying various properties.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arcs in a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arcs in a Circle” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–24.  These exercises will provide you with the opportunity to recognize various parts of a circle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Arcs in a Circle Solutions” (PDF).

      This activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 8.4 Chords in Circles  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Chords in a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Chords in a Circle” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about chords in a circle.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” (omitting Guided Practice Problem #3) sections.  This section provides examples of recognizing chords of a circle and applying various properties.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Chords in a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Chords in a Circle” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15 and 18–20.  These exercises will provide you with the opportunity to apply theorems related to chords.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Chords in a Circle Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 8.5 Inscribed Angles  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Inscribed Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Inscribed Angles” (HTML)

      Instructions: Please click on the link above and study the “Inscribed Angles” section (Pages 582–585 in the textbook), omitting the “Proof” (Pages 582 and 583) and stopping at “Review Questions.”  This section provides examples of inscribed angles in circles and their properties.

      Reading this section should take approximately 15 minutes.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Inscribed Angles”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Inscribed Angles” (HTML)

      Instructions: Please click on the link above and complete questions 1–3, 8, and 9 (“Lesson Exercises” Pages 585–588 of the textbook.  These exercises will provide you with the opportunity to apply properties of inscribed angles.  The solutions to these problems are located in the “Answers” section (starting on page 588 of the textbook).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • 8.6 Angles of Chords, Secants, and Tangents  
  • 8.7 Segments of Chords, Secants, and Tangents  
  • Unit 9: Perimeter and Area  

    Two of the most common applications of Geometry are area and perimeter.  Owning a home gives you plenty of opportunities to use these concepts.  Thinking of buying a cordless electric lawnmower?  The manufacturer will give you an estimate of the mower’s battery life using square feet.  If your yard is extremely large and your mower’s battery life is not long enough, it might take you multiple sessions to get it all mowed.  Are you buying new carpeting for your home?  It is sold in square feet or square yards.  You definitely do not want to buy too much carpeting or too little carpeting.  You are not just going to eyeball it and make a guess.  You will want to measure everything carefully, and then order a bit more, giving yourself some leeway.

    Like area, perimeter is often used in home ownership.  If you install fencing around your yard, you are going to calculate the perimeter before you buy the fencing.  If you are forced to run wiring around the walls of a room, it is vital that you know how much wiring you will need.

    Time Advisory   show close
    Learning Outcomes   show close
  • 9.1 Perimeter and Area Basics  
  • 9.2 Area of Triangles  
  • 9.3 Area and Perimeter of Rectangles and Squares  
  • 9.4 Area of a Parallelogram  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Parallelogram”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Parallelogram” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the video to learn about the area of a parallelogram.  You may omit watching the “Area of Parallelograms” video.  Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the area of various parallelograms.

      Reading the material, watching the video, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Parallelogram”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Parallelogram” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–16.  These exercises will provide you with the opportunity to apply the formulas for the area of a parallelogram.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Area of a Parallelogram Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 9.5 Area and Perimeter of Trapezoids  
  • 9.6 Area and Perimeter of Rhombuses and Kites  
  • 9.7 Area of Similar Polygons  
  • 9.8 Circumference  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Circumference”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Circumference” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the circumference of a circle.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the circumference of a circle.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Circumference”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Circumference” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–18.  These exercises will provide you with the opportunity to apply the formulas for circumference of a circle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Circumference Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 9.9 Arc Length  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arc Length”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arc Length” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the arc length of a circle.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the arc length of a circle.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arc Length”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Arc Length” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to apply the formulas for arc length of a circle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Arc Length Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 9.10 Area of a Circle  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Circle” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the area of a circle.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the area of a circle.

      Reading the material, watching the videos, and pausing to take notes should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Circle”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Area of a Circle” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–15.  These exercises will provide you with the opportunity to apply the formulas for area of a circle.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Area of a Circle Solutions” (PDF).

      Completing this activity should take approximately 45 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Assessment: Khan Academy’s “Area of a Circle”

      Link: Khan Academy’s “Area of a Circle” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on the area of circles.  You can review the concepts associated with these questions from subunit 9.10.  Determine the value for the problem, and enter your answer in the answer box.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy. 

      See a broken link? Please let us know!

  • 9.11 Area of Composite Shapes  
  • Unit 10: Surface Area and Volume  

    Finding the amount of space inside of an object is important, and something you will do in your everyday life.  If you try to fill a box with DVDs, to fill up a sandbox, or to fill a pool with water, you will use concept of volume.  There are more shapes than just simple boxes, so you will learn to find the amount of space inside different types of three-dimensional objects.

    Finding the amount of space on the outside of an object is also something you will do in your everyday life.  If you decide to paint your house, you need to know how much space is going to be painted; buying too much paint would be a waste of money, and buying too little paint would be a nuisance.  A few minutes of calculating can ensure that you get the amount of paint you need. 

    Time Advisory   show close
    Learning Outcomes   show close
  • 10.1 Polyhedrons  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “The Polyhedron”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “The Polyhedron” (HTML)

      Instructions: Please click on the link above and study the “The Polyhedron” section (Pages 665–671 in the textbook), stopping at “Review Questions.”  This reading defines various terms associated with polyhedron.

      Reading this section should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “The Polyhedron”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “The Polyhedron” (HTML)

      Instructions: Please click on the link above and complete questions 1–10 (“Lesson Exercises” Pages 671–673 of the textbook).  These questions will provide you with the opportunity to apply properties of three-dimensional figures.  The solutions to these problems are located in the “Answers” section (starting on page 673 of the textbook).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.2 Representing Solids  
    • Reading: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Representing Solids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Representing Solids” (HTML)

      Instructions: Please click on the link above and study the “The Polyhedron” section (Pages 674–681 in the textbook), stopping at “Review questions.”  This reading defines various terms and methods associated with representing solids.

      Reading this section should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Representing Solids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Representing Solids” (HTML)

      Instructions: Please click on the link above and complete questions 1–10 (“Lesson Exercises Pages 681–683 in the textbook).  These questions will provide you with the opportunity to apply properties of representing solids.  The solutions to these problems are located in the “Answers” section (starting on page 683 of the textbook).

      Completing this activity should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.3 Prisms  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Prisms”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Prisms” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of prisms.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various prisms.

      Reading these sections, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Prisms”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Prisms” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–20.  These exercises will provide you with the opportunity to apply the formulas for surface area and volumes of prisms.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Prisms Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.4 Cylinders  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cylinders”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cylinders” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of cylinders.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various cylinders.

      Reading these sections, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cylinders”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cylinders” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–16.  These exercises will provide you with the opportunity to apply the formulas to find the surface areas and volumes of cylinders.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Cylinders Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.5 Pyramids  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Pyramids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Pyramids” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of pyramids.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various pyramids.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Pyramids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Pyramids” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–19.  These exercises will provide you with the opportunity to apply the formulas to find the surface areas and volumes of pyramids.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Pyramids Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.6 Cones  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cones”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cones” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of cones.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various cones.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cones”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Cones” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–9 and 12–16.  These exercises will provide you with the opportunity to apply the formulas to find the surface areas and volumes of cones.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Cones Solutions” (PDF).

      Completing this activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.7 Spheres  
    • Assessment: Khan Academy’s “Solid Geometry”

      Link: Khan Academy’s “Solid Geometry” (HTML)

      Instructions: Please click on the link above and complete the assessment that tests your knowledge on surface area and volume.  You can review the concepts associated with these questions with the content in subunits 10.3-10.7.  Determine if the given angle is acute, right, or obtuse using the interactive protractor.  Then, click on “Check Answer” to see if you were correct or if you need to try again.

      Completing this activity should take approximately 45 minutes.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-Share-Alike License 3.0. It is attributed to the Khan Academy. 

      See a broken link? Please let us know!

    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Spheres”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Spheres” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of spheres.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various spheres.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Spheres”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Spheres” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 2–19.  These exercises will provide you with the opportunity to apply the formulas to find the surface areas and volumes of spheres.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Spheres Solutions” (PDF).

      This activity should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 10.8 Composite Solids  
    • Lecture: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Composite Solids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Composite Solids” (HTML and YouTube)

      Instructions: Please click on the link above and take notes as you watch the videos to learn about the surface area and volume of composite solids.   Then, read the “Guidance,” “Vocabulary,” and “Guided Practice” sections.  This section provides examples of finding the surface area and volume of various composite solids.

      Reading the material, watching the videos, and pausing to take notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Activity: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Composite Solids”

      Link: CK-12 Flexbook: Victor Cifarelli et al.’s Geometry: “Composite Solids” (HTML)

      Instructions: Please click on the link above, scroll down to the “Practice” section, and complete exercises 1–11 and 13–15.  These exercises will provide you with the opportunity to apply the formulas to find the surface areas and volumes of composite solids.  You may also access the material in PDF, mobi, and ePub format by clicking on the appropriate link under “Download,” if you choose to create a free account.  Once you have completed the exercises, check your answers against the Saylor Foundation’s “Composite Solids Solutions” (PDF).

      This activity should take approximately 1 hour and 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

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  • Final Exam  

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